Preeti can do a work in the same time in which Kanchan and Sudha together can do it. If Preeti and Kanchan work together, the work can be completed in 10 days. Sudha alone needs 50 days to complete the same work, then Kanchan alone can do it in how many days?

Options

A. 30 days
B. 25 days
C. 20 days
D. 15 days

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
- Preeti can do a work in the same time as Kanchan and Sudha together.
- Preeti and Kanchan together can complete the work in 10 days.
- Sudha alone needs 50 days to complete the work.

### Step 2: Calculate Sudha's work rate
Sudha can complete the work in 50 days, so her work rate (efficiency) is:
\[ \text{Sudha's efficiency} = \frac{1 \text{ work}}{50 \text{ days}} = \frac{1}{50} \text{ work/day} \]

### Step 3: Calculate the combined work rate of Preeti and Kanchan
Preeti and Kanchan together can complete the work in 10 days, so their combined work rate is:
\[ \text{Preeti + Kanchan's efficiency} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \text{ work/day} \]

### Step 4: Set up the equation for Preeti's efficiency
Let Preeti's efficiency be \( P \). According to the problem, Preeti can do the work in the same time as Kanchan and Sudha together. Therefore:
\[ P = K + S \]
where \( K \) is Kanchan's efficiency and \( S \) is Sudha's efficiency.

### Step 5: Substitute Sudha's efficiency into the equation
We know Sudha's efficiency \( S = \frac{1}{50} \). So we can express the equation as:
\[ P = K + \frac{1}{50} \]

### Step 6: Relate Preeti's and Kanchan's efficiency
From the previous steps, we know:
\[ P + K = \frac{1}{10} \]
Substituting \( P \) from the previous equation gives:
\[ (K + \frac{1}{50}) + K = \frac{1}{10} \]
This simplifies to:
\[ 2K + \frac{1}{50} = \frac{1}{10} \]

### Step 7: Solve for Kanchan's efficiency
To isolate \( K \), first subtract \( \frac{1}{50} \) from both sides:
\[ 2K = \frac{1}{10} - \frac{1}{50} \]
Finding a common denominator (which is 50):
\[ \frac{1}{10} = \frac{5}{50} \]
So:
\[ 2K = \frac{5}{50} - \frac{1}{50} = \frac{4}{50} = \frac{2}{25} \]
Now divide both sides by 2:
\[ K = \frac{2}{25} \times \frac{1}{2} = \frac{1}{25} \text{ work/day} \]

### Step 8: Calculate the time taken by Kanchan to complete the work alone
To find the time taken by Kanchan to complete the work alone, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} \]
The total work is 1 (the whole job), so:
\[ \text{Time} = \frac{1}{\frac{1}{25}} = 25 \text{ days} \]

### Final Answer
Kanchan alone can complete the work in **25 days**.

---